Introduction to Finding Tangent Lines
Understanding the concept of tangent lines and their equations is crucial in calculus and geometry. A tangent line to a curve at a given point is a line that just touches the curve at that point and has the same slope as the curve at that point. This article will guide you through the process of finding the equation of the tangent line to the curve (y x^2/x) at the point (x 13).
Understanding the Curve (y x^2/x)
The given function is (y x^2/x). Simplifying this expression, we get (y x), which is a simple linear function. However, for the purpose of finding the tangent line, we will use the given form (y x^2/x) to demonstrate the process.
Finding the Derivative
The first step in finding the tangent line is to determine the derivative of the function, which represents the slope of the tangent line at any point on the curve.
[y x^2/x] Taking the derivative with respect to (x), we get:
[frac{dy}{dx} 1 - frac{2}{x^2}]
Identifying the Slope at the Point of Tangency
The point of tangency given is (x 13). To find the slope of the tangent line at this point, we substitute (x 13) into the derivative:
[m frac{dy}{dx} 1 - frac{2}{13^2} 1 - frac{2}{169} frac{169 - 2}{169} frac{167}{169}]
Therefore, the slope (m) at (x 13) is approximately (0.99438202).
Formulating the Equation of the Tangent Line
The equation of a line given its slope (m) and a point ((x_1, y_1)) through which the line passes is:
[y - y_1 m(x - x_1)]
The given point is (x 13) and (y 13^2/13 13), so ((x_1, y_1)) is ((13, 13)). Substituting (m 0.99438202) and ((x_1, y_1) (13, 13)) into the equation, we get:
[y - 13 0.99438202(x - 13)]
Rearranging the terms, we find:
[y 0.99438202x - 12.92686626 13]
[y 0.99438202x 0.07313374]
Conclusion and Visualization
The equation of the tangent line to the curve (y x^2/x) at the point (x 13) is approximately (y 0.99438202x 0.07313374). A plot of this function and the tangent line would show the line closely approximating the curve near the point of tangency.